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The Tunneling Percolation Staircase
Isaac Balberg
Racah Institute of Physics,
The Hebrew University, Jerusalem, Israel
Stat. Mech. Day IV. 230611

GRANULAR METAL COMPOSITES
σ (xxc)t
t = 1.9

Thermal switches
Airplane wheels
Self regulated
heaters
Papyrus glyphs
CB
CB polymer composites and applications

Two fundamental questions
1) What determines the conductivity threshold?
2) What determines the shape of the σ(Φ) dependence?
Phys. Rev. B, 30, 3933 (1984): 304 Q

Confirmation of universality
0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.2410
4
103
102
101
100
(
cm
)1
CB volume fraction
t=1.95 , xc=0.10
XC72 2011
t = 1.7
tun (D = 3) ≈ 2
t
cpp )(

Experimentally observed non universal exponents
(ppc)t
t = 6.4
vc = 39 vol.%
p = v/vc
R (≥ R0L1)
RL= R(L/)/(L/)D1
R0
(ppc)
(ppc)(ttun)

5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r (arb.units)
h(r
) or
g(r
)
g(r) = g0exp[2r/ξ]
h(r) = (1/d)exp[r/d]
f(g) g(ξ/2d1)
The diverging normalized distribution
in the tunneling percolation problem
f(g) =h(r)(dr/dg)
0gpc←p
d
ξ
t = tun+(2d/ξ)1Phys. Rev. Lett, 59, 1305 (1987): 215 Q
σ (ppc)t

Various carbon blacks
t = 6.4 t = 4 t =1.8
10 wt. %39 vol. %

Percolation and Tunneling?
V V
g(rij) = g0 exp[2(rijD)/ξ] What is the meaning of vc?
Carbon, 40, 139 (2002): 100 Q.

The hopping model
(the CPA network)
rc
Bc=(4π/3)[rc3 (2b)3]N
The increase
of N
conserves
topology but
increase the
value of g
Local
connectivity
independent of N
g(rc) = g0exp[2(rc2b)/ξ]
Phys. Rev. B, 81, 155434 (2010)

The prediction of the percolationthreshold (critical
resistor, hopping like) model and its relation to the
criticalpercolation (phasetransition like) model
ℓn(N)
ℓn(σ)
Δσc
ℓn(σ) = (aβ/ξ)Φβ+ℓn(A2)
Φ N1/3

The origin of the staircase model
p
Log(σ)
pi,
Zi
g(rij) = g0exp[2(rij)/ξ]
J. Phys. D: 42, 064003 (2009)
pc→ Bc/Z

The lattice and continuum
composite models
D
ξ

The effect of tunneling in lattices and the continuum
Phys. Rev. B, 82, 134201 (2010)

Staircase observed in N990/Polyethylene Composites
0.45 0.50 0.55 0.60
105
104
103
102
101
(
cm)
1
CB volume fraction
t=1.90 , xc=0.415
t=1.70 , xc=0.465
t=2.05 , xc=0.500
t=1.95 , xc=0.570

Conclusions• The observed σ(Φ) behavior is a series of percolation
transitions, the stairs.
• In the lattice these have to do with the series of lattice neighbors. In the continuum these have to do with shells of neighbors that result from an RDF with peaks.
• In the continuum the σ(Φ) behavior of each stair is that of a universal behavior while the envelope of the stairs yields a non universal percolation behavior which simply presents hopping.
• The experimentally observed Φc is either of a “universal” stair or a result of the fact that the data do not go to Φ = 0. Percolation is well confirmed if both, a universal behavior and a non hopping behavior are exhibited by the data.
• One can get a simple universal percolation behavior if all the resistors are nearly equal (non divergent distribution) and all belong to a random network (as for anisotropic particles).

The End

RF transmission (“AC Conductivity”) at 80 MHz
0.45 0.50 0.55 0.6010
5
104
103
102
101
100
(
cm
)1
CB volume fraction
t=0.4 , xc=0.415
t=0.7 , xc=0.465
t=1.65 , xc=0.500
t=0.45 , xc=0.570

f(g
)
g
Area = p = 2pcArea = pc
What if we have a distribution of (or r0 = 1/g) values?
RL (ppc)tun
= gc∫1(1/g)f(g)dg.
p[gc∫1f(g)dg] = pc
f(g) = (1)g yields that:gc = [(ppc)/p]
1/(1)
(ppc)/(1) (ppc)
(ttun)

Universal and Hopping Interpretations in
XC72Polyethylene Composite
1.6 1.7 1.8 1.9 2.0 2.1 2.2
104
103
102
101
100
(
cm
)1
(CB volume fraction)1/3
, x1/3
(b)
0.01 0.110
4
103
102
101
100
(
cm
)1
CB , xxc
measured data
t=1.95 , Xc=0.1

NonHopping but Percolation Behavior
0.24 0.26 0.28 0.30 0.32 0.34
1010
109
108
107
106
(Si content)1 (vol. %)1
(Si content)1/3 (vol. %)1/3
0.015 0.020 0.025 0.030 0.035 0.040 0.045
Co
nd
uc
tiv
ity
(
cm
)1
Phys. Rev. B, 83, 035318 (2011)

t
cpp )(
Percolation Transition
t = tun = (D2)+
(1p)
tun (D = 2) ≈ 1.3
tun (D = 3) ≈ 2

The LNB Model of the backbone
R (≥ r0L1)RL= R (L/)/(L/)D1
RL (ppc)[ +(D2))] (ppc)
tun L1(ppc)/pc =1 L1 (ppc)1
(ppc)
ζ ≥ 1
r0L1 L
S&A 1992

The electrical conductance in composites
and the analysis of the conductance network
One expects intuitively and we can show rigorously that
VVc = (NNc)(vHC) (vol.%) in the continuum, is equivalent to pspsc in lattices

TUNNELING (Wiesendanger, 1994)
LL
L ≤ 10 nm
σ(r) = σ0exp[2r/L]
E

Resistors Equivalent Network
(Ii) = (gij)(vivj)
R = (v1vN)/I
I1 = IN = I, Ii (i # 1,N) = 0
Gii = Σgi,j, Gij = gji
(I) = G (v)
(v) = G1(I)
Phys. Rev. B, 28, 3799 (1983): 162 Q

The critical behavior of the resistance of
pencil generated linesegments system
1/gl = A0l
l
J. Water Resources Research 29, 775 (1993) : 101 Q

Non Universal and Hopping Interpretations
0.45 0.50 0.55 0.6010
6
105
104
103
102
101
100
(
cm)
1
CB (volume fraction)
t=10.95 , xc=0.305
(a)
1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34
105
104
103
102
101
(
cm
)1
(CB volume fraction)1/3
, x1/3
(b)

The model of Kogut and Straley
f(g) = (1)g
• =gc∫1 [f(g)/g]dg (gc
 1), gc = [(ppc)/p]1/(1)
• we have here the three types of = F(gc)behaviors; stronglydiverges for 0 < < 1, logarithmically diverges for = 0 and it will not diverge for < 0.
• Correspondingly: (ppc)/(1),
• log(ppc) and = [(1)/()] = const.

The basics of nonuniversal behavior
RL (ppc)tun
= gc∫1(1/g)f(g)dg.
p[gc∫1f(g)dg] = pc
Example (K&S, 1979): f(g) = (1)g yields that:
gc = [(ppc)/p]1/(1)
(ppc)/(1) (ppc)
(ttun)
( = [(1)/()])

“RDF” of the polymer shells

“RDF” of the polymer shells

Evidence for stairs in MWCNTpolymer composites

Some idea about the materials

Evidence for stairs in MWCNTpolymer composites

Conductivity and viscosity

Conductivity and viscosity

RDF and Cumulative coordination number

RDF in the Continuum

Statistics of the t values in CNTpolymer
composites

Two presentations of the same data
Percolation interpretation “hopping” interpretation

Non Universal and Hopping Interpretations
0.4 0.5 0.610
6
105
104
103
102
101
100
Co
nd
uc
tiv
ity
(c
m)
1
x (volume fraction)
1.15 1.20 1.25 1.30 1.3510
6
105
104
103
102
101
100
101
Co
nd
uc
tiv
ity
(c
m)
1
x1/3
(x volume fraction)
1.6 1.4 1.2 1.0 0.8 0.610
6
105
104
103
102
101
100
Co
nd
uc
tiv
ity
(c
m)
1
log10
(x  xC)

The physical and mathematical resemblance of the
percolation and (CPA) hopping behaviors:
In the classical hopping (CPA) model :log σ = δc/ξ+log A1 = (aβ/ξ)x
β +log A1 1/3 ≤ β ≤ 1
In our percolation model: log σ = t log(xxc) +log A2and we saw (from the Excluded volume argument) that
t = tun+d/ξ 1 so that for t >> tun we have that t ≈ (xβ)/ξ
where 1/3 ≤ β ≤ 1 and log (xxc) < 0.
• The two models yield then quite a similar dependence on x. Experimentally; since usually x >> xc these two are usually non distinguishable. Can we still distinguish between them?

Presentation of the CPA results in the case of spheres

CBpolymer composites
•Non universal value of t ≈ 6 for low structured CB composites
(ppc)t
t = 6.4
vc = 39 vol.%
p = v/vc
R (≥ r0L1)
Phil. Mag. B, 56, 991 (1987): 142Q
RL= R(L/)/(L/)D1

Non Spherical Fillers

Graphene composites

effects of particle shape asymmetry: spheroids (ellipsoids of revolution)
GTN for colloidal conductorinsulator composites
prolate spheroid (a/b >1)
oblate spheroid (a/b

σ(v) behaviors
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910
7
106
105
104
103
102
v content
2
1
Co
nd
uc
tiv
ity
(
cm
)1
Ge film
NiSiO2

Typical behavior of all composites
Percolation Threshold

Non Universal behavior

Geometry of Local Resistors
RV
IRVlike
(microemulsions)
PT
Local
connectivity
criterion?
IRV
(porous media,
impurities in SC
model)

A percolation threshold (critical resistor)
system vs. a critical percolation (phase
transition like) system
ℓn(N)
ℓn(σ)
Δσc
log σ = (aβ/ξ)xβ+log A2

Critical Path AnalysisOne labels the conductances (the interparticle
distances) in the system in descending order
until one gets percolation i.e., one finds the
thinnest soft shell δ/2 that still provides
percolation for a given Φ.
Actually, we implant hard core particles and
then attach to each a shell of thickness δ/2.
We check then how many particles, or how
high a Φ is needed in order to get a
percolation path.
The agreement of the CPA with simulation of
the resistors network suggests that one finds
the largest σ(Φ) without having to calculate
the entire resistor networks.
This works because of
the very wide resistor
distribution values

The effect of tunneling in lattices and the continuum

The tunneling percolation staircase
p
Log(σ)
pi,
Zi
tun= 1.9
J. Mod. Phys. B, 18, 2091 (2004): 52 Q

The tunneling percolation staircase
p
Log(σ)
pi,
Zi
tun= 1.9
Int. J. Mod. Phys. B, 18, 2091 (2004): 52 Q

Two interpretations of the full set of data points

LATTICE PERCOLATION
ps
pb
S
pc
r0
J. Water Resources Research 29, 775 (1993) : 101 Q

A typical continuum system
No p here
Local
connectivity
by partial
overlap

The End

The Staircase in the AC measurements
0.45 0.50 0.55 0.60
108
109
1010
Y A
xis
Title
X Axis Title
C
t=0.4
t=0.7
t=1.65
t=0.45

Fluctuations in “experimental “ data

Two presentations of the same data

Two presentations of the same data

The volume and excluded volume
of the capped cylinders
v = (4/3)(W/2)3+(W/2)2L
= (32/3)(W/2)3+8π(W/2)2L+4(W/2)L2
= /4
Nc = (Bc/v)v/ (Bc/v)(W/L)
when L

ElectroThermal Switching: the „global‟ effect
• an increase of the macroscopic resistance above a threshold bias
• a stationary “switched” state is reached within 30 sec
20 10 0 10 200.3
0.2
0.1
0.0
0.1
0.2
0.3
Cu
rre
nt
(A)
Voltage (V)
t~1 sec
t=30 and t=300 sec
Phys. Rev. Lett. 90, 236601 (2003): 20 Q.
V=3
V
V=6 V V=8 V

Making the Audio Disc

HISTORY
• 1982 RCA Laboratories, Princeton, USA
• Request: High conductivity, High elasticity.
• Question: Carbon Black supplier?
• A) Cabot Corporation, Boston USA
• B) Akzo Chemicals, Amersfoort, The Netherlands

Low structure or high structure ?

Video Disc Radius (In)
The “lost” signal (Percolation Threshold and Anisotropy)
Phil Mag. B, 56, 991 (1987): 142 Q.

Overcoming the “Anisotropy”
The Question is:
How to express
conceptually and
mathematically the
feeling that
elongated, not
parallel, particles will
yield a higher
conductivity (i.e.
connectivity) for the
same CB content?

Cylinder like objects
Phys. Rev. B, 28, 3799 (1983): 156 Q

Behavior of three types of composites that are
based on elongated carbon particles

Typical behavior of all composites
Percolation Threshold

Typical behavior of all composites
Percolation Threshold

the percolation picture implies the presence of a sharp cutoff, but tunneling does not
percolation versus tunneling
g(r12) = 0 for r12 > D+d
percolation
g(r12) ≠ 0 for r12 < D +d
tunneling
g(r12) = g0 exp[2(r12D)/]
tunneling does not
imply any sharp cut
off
Carbon, 40, 139 (2002): 82 Q.

The concept of “excluded volume”
R
R
2R
Vex = v2D
The Excluded
Volume is the
volume in which
the center of the
“other object”
has to be in order
for the two
objects to, at least
partially, overlap
vNc = (v/Vex)(VexNc) = (v/Vex)Bc
We generalized v2D to other objects calling it “Excluded Volume”
Note: 2D = smallest Vex/v
for permeable objectsPhys. Rev. B, 30, 3933 (1984): 290 Q
NVex = B: the average
number of overlapping
objects, i.e. the average
number of bonds per
object. Bc is expected
then to be a topological
invariant.

History of percolation Thresholds in
the Continuum (Sastry, PRE 2007)
Table 2. Key percolation studies in three dimensional random fibrous materials
Authors Year Reference Arrangement Objects Contribution Approach
Balberg, Binenbaum and
Wagner
1984 27 isotropic capped cylinders
First study of percolation of random objects in three
dimensions
Monte Carlo
simulations
Balberg, Anderson, Alexander et al.
1984 26 isotropic capped cylinders
Presented the conjecture that percolation threshold for a system of identical objects in three dimensions is inversely
proportional to excluded volume of one object
Monte Carlo simulations
Bug, Safran and
Webman 1985 28 isotropic
capped
cylinders
Confirmed excluded volume
rule and showed constant of proportionality equals one in slender rod limit
cluster
expansion
Neda, Florian and Brechet
1999 29 isotropic capped cylinders
Performed Monte Carlo simulations of 3D stick systems using correct isotropic
distribution and confirmed excluded volume theory
Monte Carlo simulations
Yi and Sastry 2004 30 isotropic ellipsoids
Presented analytical
approximation for the percolation threshold of two and three dimensional arrays of overlapping ellipsoids
series
expansion
Yi, Wang, and Sastry 2004 31
Uniform distribution in xy plane. Range of
distributions from parallel to random in zdirection
ellipsoids
Provided comparisons of cluster sizes, densities, and percolation points for two and threedimensional systems of
overlapping ellipsoids to investigate the range of applicability of a 2D model for predicting percolation in thin
threedimensional systems
Monte Carlo simulations

The generalization to the hard core caseIn considering
tunneling, the
tunneling range is
Rr = δ
The important
observation is that :
Фc = Ncv (Bc/ΔVex)v
r2L/[L2(Rr)]
[r/(Rr)](r/L)
If δ = (Rr) we
have that
Фc (r/δ)(r/L)
For L >> R
Vex ≈ 4L2R
ΔVex ≈ 4L2δ

For Oblate Particles
Asymptotically for a >> b, d and isotropic distribution:
ΔVex [(a+d)3 –a3] ≈ (a2d), v = πa2b, Фc b/d = (a/d)(b/a)
From the simulations:
Фc v/Vex ≈ (a/d)(b/a)1/2
Phys. Rev. B, 81,155434 (2010).
For discs of
radius a and
thickness b
Vex a3

Resistors Equivalent Network

The tunneling percolation staircase
p
Log(σ)
pi,
Zi
tun= 1.9

f(g
)
g

The RDF

The effect of a tunneling like distribution of conductances
t = 6.4f(g) = (1)g
(ppc)/(1)
f(g) g(ξ/d1)
t = tun+(d/ξ)1
t
cpp )( V V

Resistors Equivalent Network

f(g
)
g

f(g
)
g

A percolation threshold (critical resistor)
system vs. a critical percolation (phase
transition like) system
ℓn(N)
ℓn(σ)
Δσc
log σ = (aβ/ξ)xβ+log A2

Critical Path AnalysisOne labels the conductances in the system in
descending order until one gets percolation
i.e., one finds the largest soft shell δ that is
needed to get percolation for a given Φ.
Actually, one implants hard core particles and
then attaches with a shell δ/2. One checks
then how many particles, or how high a Φ is
needed in order to get percolation.
The agreement of the CPA with simulation of
the resistors network suggests that one finds
the largest σ(Φ) without having to calculate
resistor networks.This works
because of the
very wide
distribution of
resistor values

Presentation of the CPA results in terms of percolation
0.0 0.1 0.2 0.3 0.40
10
20
30
40
50
60
Threshold Content C
Th
e e
xp
on
en
t
t

application of CPA formulas to real conductorpolymer nanocomposites
GTN for colloidal conductorinsulator composites
from the knowledge of vs f and from the filler geometry [a/b and
D=2max(a,b)] one can estimate the value of the tunneling factor

The physical and mathematical resemblance of the
percolation and (CPA) hopping behaviors:
In the classical hopping (CPA) model :log σ = d/ξ = (aβ/ξ)x
β +log A1 1/3 ≤ β ≤ 1
In our percolation model: log σ = t log(xxc) +log A2we saw (from the Excluded volume argument) that
t = tun+d/ξ 1 so that for t >> tun we have that t ≈ xβ/ξ
where 1/3 ≤ β ≤ 1 and log (xxc) < 0.
• The two dependence yield then quite a similar dependence on x. Experimentally; since usually x >> xc these two are usually non distinguishable. Can we still distinguish between them?

application of CPA formulas to real conductorpolymer nanocomposites
GTN for colloidal conductorinsulator composites
even if a/b changes over 6 orders of magnitude, the tunneling factor is found to be within the
expected range 0.1 nm < < 10 nm [G. Ambrosetti et al. Phys. Rev. B 81, 155434 (2010)]

The effect of tunneling in lattices and the continuum

Analysis of a resistors network

Critical Path AnalysisOne labels the conductances in the system in
descending order until one gets percolation.
One takes then the smallest conductance thus found
and determines its δ(Φ) value. The agreement
with simulation suggests that:
One can simply arrange the δ(Φ) values and
find directly the largest needed δ(Φ)
without having to calculate resistor
networks.
This works because of the very wide
distribution of resistor values

The excluded volume approach• The parameter that is known experimentally is the critical volume % of the “filler” Φc, so
the prediction to be made is how does Φc depend on the microscopic parameters.
• For spheres these parameters are D and so:
• v= (4π/3)(D/2)3, Φ = Nv, Φc = Ncv,
• Bc = (4π/3)(rc3D3)N and δc = rcD
• Case 1: rc >> D → Bc = (4π/3)(δc3)N = (4π/3)(δc
3)Φ/v
• → δc (Φ/v)1/3 →
• Log ρ = α1 + β1Φ1/3.
• Case 2: rc ≈ D → Bc ≈ (4π/3)(3δcD2)N = (4π/3)(3δcD
3)N/D = (3δc8)Φ/D
• Φ/v → Bc = 24 δcΦ/D → δc = BcD/24Φ
• δc DΦ1
• Log ρ = α1 + β1Φ1.

The predicted CPA conductivity
dependence on Φ
• For capped cylinders:
• v ≈ πR2L, Φ = Nv = πR2LN
• ΔVex ≈ πL2δc, NΔVex = Bc
• δc ≈ BcR[(R/L)]Φ1 Φ1
• In general then, δc Φβ where in 3D: 1/3 ≤ β ≤ 1

CPA for spheres

The RDF

The End

VISCOSITY vs. CONDUCTIVITY

The differences between the different
types of elongated Carbon particles
For high structure
carbon black the
aspect ratio is of the
order of 10

The End

Recent results of the RF conductivity and
permitivity of Carbon Black composites

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.47.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
measured 0.1GHz
measured 1GHz
measured 10GHz
calculated 0.1GHz
calculated 1GHz
calculated 10GHz
log
10(
(s
ec
1))
log10
((ppc)/p
c)

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.40.8
1.2
1.6
2.0
2.4
2.8
3.2
measured 0.1GHz
measured 1GHz
measured 10GHz
calculated 0.1GHz
calculated 1GHz
calculated 10GHz
log
10(
)
log10
((ppc)/p
c)

Conductivity exponent
0 2 4 6 8 10 12 14 16 18
1.0
1.5
2.0
2.5
3.0
3.5
t
f (GHz)
N990
Raven
XC72

Permitivity exponent
0 2 4 6 8 10 12 14 16 180.0
0.4
0.8
1.2
1.6
2.0
N990
Raven
XC72
s
f (GHz)

f(g
)
g

An illustration of the basic answer
Occupied sites
Empty sites
If resistors R3 >> R2 >> R1 are attached, the resultant resistance of
the system will be dominated by R2, i.e., for a given p, an R2 resistor
will yield a threshold resistance of the system (R1→ 0 and R3 → ∞).
Approach: Keeping p and going from Z1 to Z2 in the square lattice
Bc = zpc =pzc

The staircase in a lattice
p
Log(σ)
pi, Zi

The change of the average local
resistance upon threshold approach
(the bypassing network)
• 1) Start from p > pc• 2) Consider the g‟s in descending order
• 3) The largest pc of them yield percolation
• 4) The smallest g in this subset is gc.
• 5) If there is a distribution, gc decreases as p → pc.
• 6) At p = pc the resistor of smallest g must be included
• 7) If it is g → 0 it may result that → ∞
UPON p→pc NOT ONLY THE NETWORK SHRINKS BUT THE AVERAGE RESISTANCE INCREASES AND MAY DIVERGE

A typical (pencil generated)
continuum system
No p
no f.
and N is not
informative
Phys. Rev. B, 28, 3799 (1983): 156 Q

The lattice composite model

Statistics and conductance of a composite system

RDF of spherical particles

Staircase structure in nano Graphite composites

Non Universal and Hopping Interpretations
0.45 0.50 0.55 0.6010
6
105
104
103
102
101
100
(
cm)
1
CB (volume fraction)
t=10.95 , xc=0.305
(a)
1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34
105
104
103
102
101
(
cm
)1
(CB volume fraction)1/3
, x1/3
(b)

NiSiO2
104
103
102
101
100
0.20 0.30 0.40 0.50
104
103
102
101
100
Co
nd
uc
tiv
ity
(c
m)
1
x (volume fraction)
t = 3.3

Two fundamental questions
1) What determines the
percolation threshold?
2) What determines the
shape of the σ(x)
dependence?

Another a priori unexpected behaviors
0.4 0.5 0.6 0.7 0.8 0.910
1
100
101
102
vol. fraction (v)
2
1
Co
nd
uc
tiv
ity
(
cm
)1
fractional Ni content

Percolation and tunneling percolation

Various carbon blacks

Two Conclusions The understanding of the conductivity dependence
on the content of the carbon filler in terms of
percolation theory is based on:
σ (ΦΦc)t
• 1) The larger the aspect ratio and the isotropy
the smaller the percolation threshold (the
smaller the content of carbon particles needed
for the onset of large conductivity), the
“excluded volume” argument.
• 2) The Behavior of the conductivity is
determined by the distribution of the tunneling
conductances (or interparticle distances) which
is manifested by the deviation of the critical
exponent t from its universal value.

The origin of the staircase model
p
Log(σ)
pi,
Zi
g(rij) = g0exp[2(rij)/ξ]
J. Phys. D: 42, 064003 (2009)
pc→ Bc/Z

The NLB Model
A model for the links
R (≥ r0L1)RL= R(L/)/(L/)D1
(ppc)[ +(D2))] (ppc)
tun(L1 (ppc)
1)

An illustration of a Graphene composite
v = πR2d
Vex = π2R3
v/Vex d/R

The Graphenecomposite problem
• For example, for high density spheres
• δc = BcD/24,
• for cylinders δc ≈ BcR[(R/L)]Φ1 and
• for discs Bc = π2(3b2δ+3bδ2+δ3)N so that
• δ/t ≈ Bc/(3πФ), or ≈ Ф ≈ [t/5δ)], or . δ ≈ tBc/(3πФ),